Fundamental form IV and curvature formulas of the hypersphere

Erhan GÜLER

doi:10.26637/MJM0804/0116

Abstract

We study curvature formulas and the fourth fundamental form IV of hypersurfaces in the four dimensional Euclidean geometry $\mathbb{E}^4$. We calculate fourth fundamental form and curvatures for hypersurfaces, and also for hypersphere. Moreover, we give some relations of fundamentals forms, and curvatures of hypersphere.

Keywords:

Euclidean spaces, four space, hypersurface, hypersphere, curvature, fourth fundamental form.

Mathematics Subject Classification:

Mathematics

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Erhan GÜLER Bartın University, Faculty of Sciences, Department of Mathematics 74100 Bartın, Turkey. https://orcid.org/0000-0003-3264-6239

Pages: 2008-2011
Date Published: 01-10-2020
Vol. 8 No. 04 (2020): Malaya Journal of Matematik (MJM)

Alias, L.J., Gürbüz, N.: An extension of Takashi theorem for the linearized operators of the highest order mean curvatures, Geom. Dedicata 121, 113-127 (2006).

Arslan, K., Bayram, B.K., Bulca, B., Kim, Y.H., Murathan, C., Öztürk, G.: Vranceanu surface in $mathbb{E}^4$ with pointwise 1-type Gauss map. Indian J. Pure Appl. Math. 42(1), 41-51 (2011).

Arslan, K., Milousheva, V.: Meridian surfaces of elliptic or hyperbolic type with pointwise 1-type Gauss map in Minkowski 4-space. Taiwanese J. Math. 20(2) 311-332 (2016).

Arvanitoyeorgos, A., Kaimakamis, G., Magid, M.: Lorentz hypersurfaces in $mathbb{E}_1^4$ satisfying $Delta H=alpha H$. Illinois J. Math. 53(2), 581-590 (2009).

Barros, M., Chen, B.Y.: Stationary 2-type surfaces in a hypersphere. J. Math. Soc. Japan 39(4), 627-648 (1987).

Barros, M., Garay, O.J.: 2-type surfaces in $S^3$. Geom. Dedicata 24(3), 329-336 (1987).

Bektaş, B.; Canfes, E.Ö; Dursun, U.: Classification of surfaces in a pseudo-sphere with 2-type pseudo-spherical Gauss map. Math. Nachr. 290(16), 2512-2523 (2017).

Chen, B. Y.: On submanifolds of finite type. Soochow J. Math. 9, 65-81 (1983).

Chen, B.Y.: Total mean curvature and submanifolds of finite type. World Scientific, Singapore (1984).

Chen, B.Y.: Finite type submanifolds and generalizations. University of Rome, 1985.

Chen, B.Y.: Finite type submanifolds in pseudoEuclidean spaces and applications. Kodai Math. J. 8(3), 358-374 (1985).

Chen, B.Y., Piccinni, P.: Submanifolds with finite type Gauss map. Bull. Austral. Math. Soc. 35, 161-186 (1987).

Cheng, Q.M., Wan, Q.R.: Complete hypersurfaces of $mathbb{R}^4$ with constant mean curvature. Monatsh. Math. 118, 171-204 (1994).

Cheng, S.Y., Yau, S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195-204 (1977).

Choi, M., Kim, Y.H.: Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map. Bull. Korean Math. Soc. 38 , 753-761 (2001).

Dillen, F., Pas, J., Verstraelen, L.: On surfaces of finite type in Euclidean 3-space. Kodai Math. J. 13, 10-21 (1990).

Do Carmo, M., Dajczer, M.: Rotation Hypersurfaces in Spaces of Constant Curvature. Trans. Amer. Math. Soc. 277, 685-709 (1983).

Dursun, U.: Hypersurfaces with pointwise 1-type Gauss map. Taiwanese J. Math. 11(5), 1407-1416 (2007).

Dursun, U., Turgay, N.C.: Space-like surfaces in Minkowski space $mathbb{E}_1^4$ with pointwise 1-type Gauss map. Ukrainian Math. J. 71(1), 64-80 (2019).

Ferrandez, A., Garay, O.J., Lucas, P.: On a certain class of conformally at Euclidean hypersurfaces. In Global Analysis and Global Differential Geometry; Springer: Berlin, Germany 48-54 (1990).

Ganchev, G., Milousheva, V.: General rotational surfaces in the 4-dimensional Minkowski space. Turkish J. Math. 38, 883-895 (2014).

Garay, O.J.: On a certain class of finite type surfaces of revolution. Kodai Math. J. 11, 25-31 (1988).

Garay, O.: An extension of Takahashi's theorem. Geom. Dedicata 34, 105-112 (1990).

Güler, E., Hacisalihoğlu, H.H., Kim, Y.H.: The Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4-space. Symmetry 10(9), 1-12 (2018).

Güler, E., Magid, M., Yaylı, Y.: Laplace-Beltrami operator of a helicoidal hypersurface in four-space. J. Geom. Symm. Phys. 41, 77-95 (2016).

Güler, E., Turgay, N.C.: Cheng-Yau operator and Gauss map of rotational hypersurfaces in 4-space. Mediterr. J. Math. 16(3), 1-16 (2019).

Hasanis, Th., Vlachos, Th.: Hypersurfaces in $mathbb{E}^4$ with harmonic mean curvature vector field. Math. Nachr. 172, 145-169 (1995).

Kim, D.S., Kim, J.R., Kim, Y.H.: Cheng-Yau operator and Gauss map of surfaces of revolution. Bull. Malays. Math. Sci. Soc. 39(4), 1319-1327 (2016).

Kim, Y.H., Turgay, N.C.: Surfaces in $mathbb{E}^4$ with $L_1$ pointwise 1-type Gauss map. Bull. Korean Math. Soc. 50(3), 935-949 (2013).

Kühnel, W.: Differential geometry. Curves-surfacesmanifolds. Third ed. Translated from the 2013 German ed. AMS, Providence, RI, 2015.

Levi-Civita, T.: Famiglie di superficieisoparametrichenellordinariospacioeuclideo. Rend. Acad. Lincei 26, 355-362 (1937).

Moore, C.: Surfaces of rotation in a space of four dimensions. Ann. Math. 21, 81-93 (1919).

Moore, C.: Rotation surfaces of constant curvature in space of four dimensions. Bull. Amer. Math. Soc. 26, 454-460 (1920).

Senoussi, B., Bekkar, M.: Helicoidal surfaces with $Delta^J r=$ $A r$ in 3-dimensional Euclidean space. Stud. Univ. BabeşBolyai Math. 60(3), 437-448 (2015).

Stamatakis, S., Zoubi, H.: Surfaces of revolution satisfying $Delta^{I I I} x=A x$. J. Geom. Graph. 14(2), 181-186 (2010).

Takahashi, T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan 18, 380-385 (1966).

Turgay, N.C.: Some classifications of Lorentzian surfaces with finite type Gauss map in the Minkowski 4-space. J. Aust. Math. Soc. 99(3), 415-427 (2015).